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\title{Mathematical Writing \\ Chapter 9: Writing and Defending a Thesis}
%(1.1-1.2) 
%\institute{上海立信会计金融学院}
\author{NJH}
%\date{}

\maketitle

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\begin{frame}{Contents }

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\begin{myenumerate}
\item[9.a.] Quotations
\item[9.b.] Introduction
\item[9.1.] The Purpose of a Thesis
\item[9.2.] Content
\item[9.3.] Presentation
\item[9.4.] The Thesis Defence
%\item[9.c.] Oral Examination Procedure
\item[9.5.] Further Reading
\end{myenumerate}

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\begin{frame}{9.a.1. Quotations }

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\begin{myitemize}
%\item  1987: Student writes Ph.D. thesis completely in verbatim environment.\\
%
%\hfill {\footnotesize David F. Griffiths and Desmond J. Higham}\\
%%- DAVID F. GRIFFITHS and DESMOND J. HIGHAM, 
%\hfill {\footnotesize Great Moments in \LaTeX \,\, History (1997)}

\item  {\color{red}Remember to begin by writing the easiest parts first ...} \\
It is surprising how many people believe that a thesis ... \\
should be written in the order that it will be printed and subsequently read. 

\hfill {\footnotesize Estellee M. Phillips and D. S. Pugh, How to Get a PhD (1994)}
%- ESTELLE M. PHILLIPS and D. s. PUGH, How to Get a PhD (1994)

\hspace{0.5cm}

\item  {\color{blue}Calvin: I think we've got enough information now, don't you?\\
Hobbes: All we have is one ``fact'' you made up.\\
Calvin: That's plenty. By the time we add an introduction, a few illustrations, and a conclusion, it will look like a graduate thesis.}

\hfill {\footnotesize Calvin, Calvin and Hobbes by Bill Watterson (1991)}
%- CALVIN, Calvin and Hobbes by Bill Watterson (1991)

\end{myitemize}

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%\begin{frame}{9.a.2. Enough information for a graduate thesis }
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\begin{frame}{9.b. Introduction }

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\begin{myitemize}
\item  Virtually all that has been said in Chapters 6 and 7 about writing and revising a paper applies to theses. 

\item  {\color{red} The term ``dissertation'' is synonymous with thesis, and is preferred by some.} 

\item  In this chapter I give some specific advice that takes into account the special nature and purposes of a thesis written for an advanced degree. Much of the discussion applies to undergraduate level projects too.

\end{myitemize}

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\begin{frame}{9.1. The Purpose of a Thesis }

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\begin{myitemize}
\item  The purpose of a thesis varies with the type of degree and the institution. 

\item  {\color{red}The thesis might have to satisfy one or more of the following criteria:}

    \begin{myitemize}
    \item  {\color{red}show} that the student has read and understood a body of research literature,
    \item  {\color{red}provide} evidence that the student is capable of carrying out original research,
    \item  {\color{red}show} that the student has carried out original research, and 
    \item  {\color{red}represent} a significant contribution to the field.
    \end{myitemize}

\item  It is worth checking what is expected by your institution.

\end{myitemize}

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\begin{frame}{9.2.1. Content - 1 }

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\begin{myitemize}
\item  {\color{red}A thesis differs from a paper in several ways.}
    \begin{myenumerate}
    \item  {\color{red}A thesis} must be self-contained. Whereas a paper may direct the reader to another reference for details of a method, experimental results, or further analysis of a problem, a thesis must stand on its own as a complete account of the author's work on the subject of investigation.
    
    \item  {\color{red}A thesis} is formatted like a book, broken into chapters rather than sections.
    
    \item  {\color{red}A thesis} may include more than one topic, whereas a paper usually focuses on one.
    
    \item  {\color{red}A thesis} is usually longer than an average paper, making good organization particularly important.
    
    \item  The primary readers of {\color{red}a thesis} (possibly the only readers) are its judges, and they will read it with at least as much care as do the referees of a paper.
    \end{myenumerate}
\end{myitemize}
    
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\begin{myitemize}
\item  Since there is less pressure to save space than when writing a paper for a journal, you should generally {\color{red}include details in a thesis}. 

\item  It is important to {\color{red}demonstrate understanding of the subject}, and phrases such as "it is easily shown that" and "we omit the proof" used in the presentation of original results may seem suspicious when you have no track record in the subject.  
%(the examiners may, of course, ask for such gaps to be filled during the oral examination). 

\item  Trying to {\color{red}anticipate the examiners' questions} should help you to decide what and how much to say on each topic.

\end{myitemize}

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\begin{myitemize}
\item  The thesis should not be padded with {\color{red}unnecessary material}, %(many theses are too long), 
but results that would not normally be published can be included, perhaps in an appendix, either because they might be of use to future workers or because you might want to refer to them in a paper based on the thesis. 

\item  There is no {\color{red}``ideal'' number of pages} beyond which a thesis gains respectability, and indeed there is great variation in the length of theses among different subjects and even within a subject. 
%The supervisor (UK) or thesis advisor (US) can offer advice about the suitable length.

\item  A thesis has a fairly rigid structure. {\color{red}In the first one or two chapters the problem being addressed must be clearly described and put into context.}


\end{myitemize}

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\begin{myitemize}
\item  You are expected to demonstrate a sound knowledge and {\color{red}understanding of the existing work} on the topic by providing a critical survey of the relevant literature. 

%\item  If there is more than one possible approach to the problem, the choice of method must be justified. 

\item  For a computational project the method developed or investigated in the thesis would normally be compared
experimentally with the {\color{red}major alternatives}.

\item  At the end of the thesis, {\color{red}conclusions must be carefully drawn} and the overall contribution of the thesis assessed. 

\item  It is a good idea to {\color{red}identify open problems} and future directions for research, since being able to do so is one of the attributes required of a researcher. 

\end{myitemize}

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\begin{myitemize}
%\item  Note that a thesis does not necessarily have to present major new or improved results; in many cases the key requirement is the development and communication of original ideas using sound techniques.

\item  When you write a thesis you are usually relatively inexperienced at technical writing, so it is important to {\color{red}avoid inadvertently committing plagiarism}. % (see 36.16). 

\item  If you copy text word for word from another source you must put it in quotation marks and cite the source. 

\item  If you find yourself copying, or paraphrasing, someone else's proof of a theorem, ask yourself if you need to give the proof - 

\begin{center}
{\color{red}if it is not your own work, will it add anything to the thesis? }
\end{center}

\item  Examiners will be particularly alert to the possibility of plagiarism, so be careful to avoid committing this sin.

\end{myitemize}

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\begin{frame}{9.3.1. Presentation - 1 }

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\begin{myitemize}
\item  Each institution has rules about the presentation of a thesis. 

\item  {\color{red}Page}, {\color{red}font} and {\color{red}margin sizes}, {\color{red}line spacing} (often required to be double or one and a half times the standard spacing) and {\color{red}the form of binding} may all be tightly regulated, and non-conforming theses may be rejected on submission. 

\item  The opening pages will be required to follow a standard format, typically comprising the following items (some of which will be optional).

\end{myitemize}

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\begin{myitemize}

    \item  {\color{red}A title page}, listing the author, title, department, type of degree, and year (and possibly month) of submission.
    \item  A declaration that the work has not been used in another degree submission.
    \item  {\color{red}A statement on copyright and the ownership} of intellectual property rights.
    \item  A list of notation.
    \item  A brief statement of the author's research career.
    \item  {\color{red}Acknowledgements and dedications}.
    \item  {\color{red}Table of contents}.
    \item  List of figures.
    \item  List of tables.
    \item  {\color{red}Abstract}. 

\end{myitemize}

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\begin{myitemize}
\item  The abstract may need to be repeated on a separate application form. Once the degree has been obtained, the abstract is likely to be entered into a database such as {\color{red}\it Dissertation Abstracts International} or {\color{red}\it Index to Theses}. % (for theses from universities in Great Britain and Ireland, and available on the Web for registered users at \url{http://www.theses.com}).

\item  
%The opening pages are also the place to indicate which parts of the thesis (if any) have already been published, and which parts are joint work. 
Your library will contain previous successful theses, which you can inspect to check the required format - but bear in mind that rules of presentation can and do change. {\color{red} It is likely that a \LaTeX package will be available at your institution for typesetting theses in the official style.}

\begin{center}
\url{https://github.com/XunJM/lixinthesis}
\end{center}

\item  I recommend producing an {\color{red}index} for the thesis, although this practice is not common. 
A well-prepared index can be a significant aid to examiners and readers.


\end{myitemize}

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\begin{myitemize}
\item  {\color{red}When should you start to write the thesis? }

\begin{myitemize}
\item  My advice is to {\color{red}start sooner} rather than later. 

\item  {\color{red}In the early months of study} in which you become familiar with the problem and the literature you can begin to draft the first few chapters. 

\item  You should also start immediately to {\color{red}collect references} for the bibliography - it is difficult in the later stages to hunt for half-remembered references. 

\item  One reason for making an early start on {\color{red}writing the background and survey material} is that at this stage you will be enthusiastic - later, you may know this material so well that it seems dull and boring.

\end{myitemize}

\end{myitemize}

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\begin{myitemize}
\item  {\color{red}I encourage my students to write up their work in \LaTeX \,\, as they progress through the period of study}, so that when the time comes to produce the thesis much of the writing has been done. 

\item Since most students now typeset their own theses, this approach allows them to learn {\color{red}the typesetting system} when they are least stressed, rather than in the last hectic months. 

\item If the thesis work has progressed rapidly, they may be in the pleasant position of having one or more papers already written, upon which they can base the thesis.

%\item  Unlike a published paper, a thesis will not be read by a copy editor or proofreader. It is therefore particularly important that you thoroughly read and check the thesis before it is submitted. Your thesis advisor should read and comment on the thesis, and it is worthwhile recruiting fellow students as readers, too; even if they are not specialists in the area they should be able to offer useful suggestions for improvement.


\end{myitemize}

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%\begin{frame}{9.4. The Thesis Defence - 1 }
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%
%\begin{myitemize}
%\item  The oral defence of a thesis takes different forms in different countries. For example, in the UK the candidate answers questions posed by the examiners, but does not usually give a formal presentation, whereas in other European countries it is more common for the candidate to give a presentation followed by questions from the jury. The number of examiners also varies greatly between countries.
%
%\item  Perhaps the most important piece of advice applicable to all forms of defence is to read the thesis beforehand. The defence may take place weeks or months after you submitted the thesis, and in the meantime you may forget exactly what material you included in the thesis and where it is located. To be properly prepared you need to know the thesis inside out.
%
%\end{myitemize}
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\begin{myitemize}
\item  One of the purposes of the defence is for the examiners to satisfy themselves that {\color{red}you did the work} you claim to have done and that you understand it. 

\item As well as asking straightforward questions about the thesis, they may therefore ask you why you took the approaches you did and to justify assumptions and amplify arguments. 

\item You can also expect questions that explore your knowledge of the literature outside the immediate area of the thesis, as the examiners gauge your general familiarity with the research area.

\item  It is important that you listen to questions carefully and answer the question that is asked, not some other question. %When you are under pressure it is easy to misunderstand what the examiners ask you. 
%If you do not understand a question, say so, and the question will be repeated or rephrased.

\end{myitemize}

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\begin{myitemize}
\item  If you give a formal presentation you should aim to {\color{red}give an overview of the research area} and the work you have done and not to go too deeply into the details. 

\item  The examiners will want to see that {\color{red}you appreciate the context and significance} of your work and that you are aware of problems remaining for future research. 
%Consult Chapters 10 and 11 for practical advice on writing and giving the talk. 

\item  Finally, note that an examiner who has carefully read your thesis and attended the defence should {\color{red}know enough about you to write a reference} for your job applications. The examiners may even be able to offer advice on where to seek employment.

\end{myitemize}

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%
%\begin{myitemize}
%\item  Before beginning the examination, make it clear to the examinee that his whole professional career may turn on his performance. Stress the importance and formality of the occasion. Put him in his proper place at the outset.
%
%\item  Throw out your hardest question first. (This is very important. If your first question is sufficiently difficult or involved, he will be too
%rattled to answer subsequent questions, no matter how simple they may be.) ...
%
%\item  Every few minutes, ask him if he is nervous. . .
%
%\item  Wear dark glasses. Inscrutability is unnerving.
%
%\item  Terminate the examination by telling the examinee, ``Don't call us; we will call you.''
%
%\vfill \hfill 
%- S. D. MASON, in A Random Walk in Science (1973)
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%
%\end{myitemize}
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%\begin{frame}{9.5. Further Reading }
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%
%\begin{myitemize}
%\item   Books are available that give advice on the whole process of doing research for a Ph.D., including writing the thesis. 
%
%\item   For a UK perspective, Phillips and Pugh [226] can be recommended, while Sternberg [258] and Rudestam and Newton [241] offer a US viewpoint.
%
%\end{myitemize}
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\vspace{1cm}

\begin{center}
{\Large\color{blue}
%《数学写作》第七章: Revising a Draft
Mathematical Writing 

\vspace{0.2cm}

Chapter 10: Writing a Talk
}
\vspace{1cm}

NJH

\vspace{1cm}

November 19, 2020

\end{center}

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\begin{frame}{Contents }

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\begin{myenumerate}
\item 10.a. Quotations
\item 10.1. What Is a Talk?
\item 10.2. Designing the Talk
\item 10.3. Writing the Slides
\item 10.4. Example Slides
\item 10.5. Further Reading
%\item 
%\item 
\end{myenumerate}

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\begin{frame}{10.a. Quotations }

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\begin{myitemize}
\item {\color{red}My recommendations amount to this ...} \\
Make your lecture simple (special and concrete); \\
be sure to prove something and ask something; \\
prepare, in detail; \\
organize the content and adjust to the level of the audience; \\
keep it short, and, to be sure of doing so, \\
prepare it so as to make it flexible. 

\hfill {\footnotesize - Paul R. Halmos, How to Talk Mathematics (1974) }
%- PAUL R. HALMOS, How to Talk Mathematics (1974)

%\item 
%I always find myself obliged, 
%if my argument is of the least importance, 
%to draw up a plan of it on paper and 
%fill in the parts by recalling them to mind, 
%either by association or otherwise. 
%
%\hfill {\footnotesize - Michael Faraday}
%%- MICHAEL FARADAY

\item 
{\color{red}An awful slide} is one which contains approximately a million numbers. \\
%(and we've left our opera glasses behind). 
{\color{red}An awful lecture slide} is one which shows a complete set of engineering drawings and specifications for a super-tanker. 

\hfill {\footnotesize - Kodak Limited, Let's Stamp Out Awful Lecture Slides (1979)}
%- KODAK LIMITED, Let's Stamp Out Awful Lecture Slides (1979)

\end{myitemize}

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\begin{frame}{10.1.1. What Is a Talk?  - 1}

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\begin{myitemize}
\item %In this chapter I discuss how to write a mathematical talk. 
By talk I mean a formal presentation that is prepared in advance, such as a {\color{red}departmental seminar} or a {\color{red}conference talk}, but not one of a series of lectures to students.

\item  In most talks the speaker writes on a blackboard or displays pre-written {\color{red}transparencies} on an overhead projector. 
%(From here on I will refer to transparencies as slides, since this term is frequently used and is easier to write and say.) 

\item  I will restrict my attention to {\color{red}slides}, which are the medium of choice for most speakers at conferences, but much of what follows is applicable to the {\color{red}blackboard}. %I will assume that the speaker uses the slides as a guide and speaks freely. 

\item  {\color{red}Reading a talk word for word from the slides should be avoided}; one of the few situations where it may be necessary is if you have to give a talk in a foreign language with which you are unfamiliar. 
% (Kenny [149] offers some advice on how to do this).

\end{myitemize}

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\begin{frame}{10.1.2. What Is a Talk? - 2 }

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\begin{myitemize}
\item {\color{red}A talk has several advantages over a written paper.}
    \begin{myenumerate}
    \item[1.] Understanding can be conveyed in ways that would be considered too simplified or lacking in rigour for a journal paper.
    \item[2.] Unfinished work, or negative results that might never be published, can be described.
    \item[3.] {\color{red}Views based on personal experience are particularly effective in a talk.}
    \item[4.] Ideas, predictions and conjectures that you would hesitate to commit to paper can be explained and useful feedback obtained from the audience.
    \item[5.] {\color{red}A talk is unique to you} - no one else could give it in exactly the same way. A talk carries your personal stamp more strongly than a paper.
    \end{myenumerate}

\end{myitemize}

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\begin{frame}{10.1.3. What Is a Talk? - 3 }

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\begin{myitemize}
\item  %Given these advantages, and the way in which written information is communicated in a talk, it is not surprising that 
{\color{red}Writing slides differs from writing a paper in several respects.}

\begin{myenumerate}

\item  Less material can be covered in a talk than in a corresponding paper, and fewer details need to be given.

\item  Particular care must be taken to explain and reinforce meaning, notation and direction, for a listener is unable to pause, review what has gone before, or scan ahead to see what is coming.

\item  Some of the usual rules of writing can be ignored in the interest of rapid comprehension. For example, you can write non-sentences and use abbreviations and contractions freely.

%\item  Within reason, what you write can be imprecise and incomplete - and even incorrect. These tactics are used to simplify the content of a slide, and to avoid excessive detail. %Of course, to make sure that no confusion arises you must elaborate and explain the hidden or falsified features as you talk through the slide.

\end{myenumerate}

\end{myitemize}

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\begin{frame}{10.2.1. Designing the Talk - 1 }

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\begin{myitemize}
\item {\color{red}The first step in writing a talk is to analyse the audience. }

\begin{myenumerate}

\item  Decide what {\color{red}background material} you can assume the listeners already know and what material you will have to review. 

\item  If you misjudge the listeners' knowledge, they could find your talk incomprehensible at one extreme, or slow and boring at the other. 

\item  If you are unsure of the audience, {\color{red}prepare extra slides} that can be included or omitted depending on your impression as you go through the talk and on any questions received.

\end{myenumerate}

\end{myitemize}

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\begin{myitemize}
\item The title of your talk should not necessarily be the same as the one you would use for a paper, because your potential audience may be very different from that for a paper. To encourage non-specialists to attend the talk keep technical terms and jargon to a minimum. 

%\item I once gave a talk titled "Exploiting Fast Matrix Multiplication within the Level 3 BLAS" in a context where non-experts in my area were among the potential audience.
%
%\item I later found out that several people did not attend because they had not heard of BLAS and thought they would not gain anything from the talk, whereas the talk was designed to be understandable to them. 
%
%\item A better title would have been the more general "Exploiting Fast Matrix Multiplication in Matrix Computations" .

%\end{myitemize}
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%\begin{myitemize}
%\item  A controversial title that you would be reluctant to use for a paper may be acceptable for a talk. 
%
%\item  It will help to attract an audience and you can qualify your bold claims in the lecture. Make sure, though, that the content lives up to the title.

\item  {\color{red}It is advisable to begin with a slide containing your name and affiliation and the title of your talk.} 
%This information may not be clearly or correctly enunciated when you are introduced, and it does no harm to show it again. 

%\item  The title slide is an appropriate place to acknowledge co-authors and financial support.

%\end{myitemize}
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%\begin{myitemize}
%\item  Because of the fixed path that a listener takes through a talk, the structure of a talk is more rigid than that of a paper. 

\item  {\color{red}Most successful talks follow the time-honoured format:}
\begin{myenumerate}
\item Tell them what you are going to say.
\item Say it. 
\item Tell them what you said.
\end{myenumerate}

\end{myitemize}

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\begin{myitemize}
\item  {\color{red}At the start of the talk it is usual to outline what you are going to say}: summarize your objectives and methods, and perhaps state your conclusions. 

\item  This is often done with the aid of an overview slide but it can also be done by speaking over the title slide. 

\item  {\color{red}The aim is to give the listeners a mental road-map of the talk.} You also need to sprinkle signposts through the talk, so that the listeners know what is coming next and how far there is to go. 

%This can be done orally (example: "Now, before presenting some numerical examples and my overall conclusions, I'll indicate how the result can be generalized to a wider class of problems"), or you can break the talk into sections, each with its own title slide. 

%\item  Another useful technique is to intersperse the talk with contents slides that are identical apart from a mark that highlights the topic to be discussed next; see Figure 1 0 .1. 1 7
%
%\item Figure 10.1. Contents slide: the triangle points to the next topic.
% Introduction and Motivation
% Deriving Partitioned Algorithms
% Block L U Factorization and Matrix Inversion
% Exploiting Fast Matrix Multiplication


\end{myitemize}

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\begin{myitemize}
\item  {\color{red}Garver recommends lightening a talk by building in multiple entry points}, at any of which the listener can pick up the talk again after getting lost. An entry point might be a new topic, problem or method, or an application of an earlier result that does not require an understanding of the result's proof. 

\item  {\color{red}Multiple exit points are also worth preparing} if you are unsure about how the audience will react. They give you the option of omitting chunks of the talk without loss of continuity. 

\item  {\color{blue} A sure sign that you should exercise this option is if you see members of the audience looking at their watches, or, worse, tapping them to see if they have stopped! }


\end{myitemize}

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%\begin{myitemize}
%\item  An unusual practice worth considering is to give a printed handout to the audience. This might help the listeners to keep track of complicated definitions and results and save them taking notes, or it might give a list of references mentioned in the talk. 
%
%\item  A danger of this approach is that it may be seen as presuming the audience cannot take notes themselves and are interested enough in the work to want to take away a permanent record.
%
%\item  Handouts can, alternatively, be made available for interested persons to pick up after the talk.
%
%\item  To save space, all the boxes that surround the example slides in this chapter are made just tall enough to hold the slide's content.
%
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\begin{myitemize}
\item {\color{red}Begin the talk by stating the problem, putting it into context, and motivating it.} This initial scene-setting is particularly important since the audience may well contain people who are not experts in your area, or who are just beginning their research careers.

\item  {\color{red}The most common mistake in writing a talk is to put too much on the individual slides.} The maxim "less is more" is appropriate, because a busy, cluttered slide is hard for the audience to assimilate and may divert their attention from what you are saying. 

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\begin{myitemize}

\item Since a slide is a visual aid, it should contain the core of what you want to say, but you can fill in the details and explanations as you talk through the slide. 

\item  {\color{red} If you merely read the slide, it could be argued that you might as well not be there!}

\item  There are various recommendations about how many lines of type a slide should contain: {\color{red}a maximum of 7-8 lines is recommended by Kenny} and a more liberal 8-10 lines by Freeman et al.. 

\item  These are laudable aims, but in mathematical talks speakers often use 20 or more lines, though not always to good effect.

\end{myitemize}

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\begin{myitemize}
\item  {\color{red}A slide may be too long for two reasons: the content is too expansive and needs editing, or too many ideas are expressed. }

\begin{myitemize}

\item  Try to limit each slide to one main idea or result. More than one may confuse the audience and weaken the impact of the points you try to make. 

\item  A good habit is to put a title line at the top of each slide; if you find it hard to think of an appropriate title, the slide can probably be improved, perhaps by splitting it into two.

\item  Don't present a detailed proof of a theorem, unless it is very short. It is far better to describe the ideas behind the proof and give just an outline. %Most people go to a talk hoping to learn new ideas, and will read the paper if they want to see the details.

\end{myitemize}

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%\begin{myitemize}
%\item  When a stream of development stretches over several slides, the audience might wish to refer back to an earlier slide from a later one. To prevent this you can replicate information (an important definition or lemma, say) from one slide to another. 
%
%\item  A related technique is to build up a slide gradually, by using overlays, or simply by making each slide in a series a superset of the previous one. (The latter effect can be achieved by covering the complete slide with a sheet of paper, and gradually revealing the contents; but be warned that many people find this peek-a-boo style irritating. 
%
%\item  I do not recommend this approach, but, if you must use it, cover the slide before it goes on the projector, not after.) Overlays are best handled by taping them together along one side, and flipping each one over in turn, since otherwise precise alignment is difficult.
%
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%\begin{myitemize}
%\item  If you think you will need to refer back to an earlier slide at some particular point, insert a duplicate slide. This avoids the need to search through the pile of used slides. It is worth finding out in advance whether two projectors will be available. If so, you will have less need to replicate material because you can display two slides at a time.
%
%\item  It is imperative to number your slides, so that you can keep them in order at all times. At the end of the talk the slides will inevitably be jumbled and numbers help you to find a particular slide for redisplaying in answer to questions. I put the number, the date of preparation, and a shortened form of the title of the talk, on the header line of each slide.
%
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%每页详细内容

\begin{myitemize}
\item {\color{red}When you write a slide, aim for economy of words. Chop sentences mercilessly to leave the bare minimum that is readily comprehensible.} %Here are some illustrative examples.

\item One example: 
\begin{myitemize}
\item Original: It can be shown that $\partial \Vert Ax\Vert = \{A^T \text{dual}(Ax)\}$.
\item Shorter: Can show $\partial \Vert Ax\Vert = \{A^T \text{dual}(Ax)\}$.
\item Even shorter: $\partial \Vert Ax\Vert = \{A^T \text{dual}(Ax)\}$.
\end{myitemize}

%\item Original: The stepsize is stable if $p(z)$ is a Schur polynomial, that is, if all its roots are less than one in modulus.
%
%Shorter: Stepsize is stable if $p(z)$ is a Schur polynomial (all roots less than one in modulus) .
%
%Even shorter: Stepsize stable if $p(z)$ is Schur ($\vert \text{roots}\vert <1$).

\item Another example: 
\begin{myitemize}
\item  Original: We can perform a similar analysis for the Lagrange interpolant.
\item  Shorter: Similar analysis works for the Lagrange interpolant.
\item  Even shorter: OK for Lagrange.
\end{myitemize}

\end{myitemize}

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\begin{myitemize}


\item  {\color{red}Lists are an effective way} of presenting information concisely. Audiences like them, because there is something psychologically appealing about being given a list of things to note or remember. 
%Consider using bulleted or numbered lists of five or fewer items, expressed with parallel constructions where appropriate (see 34.16).

%\item  If you cite your own work you can reduce your name to one letter, but the names of others should not be abbreviated. Thus, from one of my slides:
%
%Original: This question answered in nonsingular case by Higham and Knight (1991).
%
%Shorter: Answered for nonsingular A by H \& Knight (1991).

\item  {\color{red}Be sure to avoid spelling mistakes} on the slides. They are much more noticeable and embarrassing in slides than in a paper.

\item  {\color{red}Graphs and pictures} help to break up the monotony of wordy slides. 
%But beware of "slide shuffling", a common mistake where a long sequence of similar slides is displayed in rapid succession ("that was the plot for $p = 0.5$, now here is $p = 0.6$, ..., and here is $p = 0.7$,. . . "). 

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\begin{myitemize}

\item  {\color{red}An audience needs time to absorb each slide} and it is difficult for anyone to remember more than the last two slides. Anything that might irritate or bore your audience is to be avoided.

\item  At the end of the talk, {\color{red}conclusions and a summary} are usually given. 
This is your last chance to impress your ideas upon the audience. 

\item  If possible, try to summarize what you have said in a way that does not presuppose understanding of the talk. 
Even if your conclusions are largely negative, try to finish the talk on a positive note! 
%The last slide is a good place to give your email address and a Web address where your relevant papers can be found.

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\begin{myitemize}
\item  {\color{red}It is of paramount importance that slides be legible when they are projected.} 
Unfortunately, slides are frequently difficult to read for people sitting towards the rear of the room.

%\item  An important point to appreciate is that many overhead projector installations exhibit keystoning, whereby the image is wider at the top than at the bottom; this is avoided if the screen is tilted forward. 
%
%\item  Keystoning causes characters at the bottom of the screen to look much smaller than those at the top.
%
%\item  Take care not to write too close to the edges of a slide, as the areas near the edges may not be clearly visible when the slide is projected. And try to leave a sizeable gap at the bottom of the slide, as the bottom of the screen may be obstructed from the view of people at the back of the audience. I recommend experimenting with a projector to determine acceptable limits.


%\end{myitemize}
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%
%\begin{myitemize}
%\item  Various recommendations have been given in the literature on the minimum height of characters, but they necessarily involve assumptions about the magnification of the original image. 
%
%\item  My advice is to experiment with a projector in an empty room of a similar size to the one that will be used for the talk-you can quickly determine how large the characters need to be. %I find that characters this large (\verb+\LARGE+ in \LaTeX at lOpt) produce a slide readable from the back of most lecture rooms.  For the comfort of the audience I prefer to use characters this large (\verb+\huge+ in \LaTeX at lOpt).

%For a given character size the font also affects readability [240, Chap. 2]. Some people prefer to use sans serif fonts for slides (fonts without small extensions at the ends of strokes): sans serif font

%
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%\begin{myitemize}
\item It is not uncommon to see slides that have been photocopied from a paper without magnification. Invariably the slides are difficult or impossible to read. {\color{red}Avoid this sin; prepare concise, legible slides specially for the talk.}

\item  The practice of photocopying from a paper is most common with tables. Better even than magnifying the original table is to rewrite it, pruning it to the information essential for that slide. 
%For example, some entries in a table of numbers may be dispensable, and it may be possible to reduce the number of significant digits. More so than in a paper, a table of numerical results may be better displayed as a graph, which is easier to absorb as long as the labels are readable.


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\begin{myitemize}
\item  The number of slides needed depends on several factors, such as the length of the talk, the amount of mathematical detail to be presented, and the number of graphs and pictures. 

\item  It also depends on the mode of presentation: a speaker who uses an overhead projector to display outline notes and who spends considerable time filling in the details on the blackboard will need far fewer slides than one who uses only the projector. 

\item  {\color{red}In my experience, it is best to allow about two minutes per slide for "slide only" talks.} %(the same recommendation is made in [86] and [102]). 

%\item  Allowing for variation in slide length, for a twenty minute talk you should aim for twelve slides or fewer, while for a fifty minute talk thirty slides is about the upper limit.


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\begin{myitemize}
\item  {\color{red}Which are better: handwritten or typeset slides?} It is a matter of personal preference. 
The best handwritten slides are just as good as the best typeset ones, and the worst of both kinds are very poor. 

\item  {\color{red}I produce all my slides in \LaTeX, for several reasons. }

\begin{myitemize}
\item  I can work from the \LaTeX source for the corresponding paper, which saves a lot of typing; typeset slides can easily be adapted and reprinted for use in other talks; 
\item  I find it easier to revise and correct typeset slides than handwritten ones, and this encourages me to produce better slides; 
\item  I have no worries about spilling liquid over my slides or losing them. % (they are never further away than an ftp call to my workstation). 
\end{myitemize}

%\item  Advantages of handwritten slides are that they can be modified right up until the start of the talk without the need for access to a computer or printer, "it is easy and inexpensive to use colours, and handwritten slides are more individual.


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%
%\begin{myitemize}
%\item  \LaTeX 2e has a document class slides [172, 35.2] designed for making slides (it supersedes S]J1EjX, a version of \LaTeX 2.09). It automatically produces large characters in a specially designed sans serif font. It supports the production of coloured slides, via the eolor package, and overlays (so that one slide can be superimposed on another). 
%
%\item  Various other macro packages are available for use with \LaTeX, including the seminar package by Timothy Van Zandt (available from CTAN-see 313.5). Note that typeset slides are easier to read if they are not right-justified and if hyphenation is turned off.
%
%\item  To break the monotony of black and white typeset slides you can underline headings and key phrases with a coloured pen. You can also leave gaps to be filled during the talk, later wiping the slides clean, ready for the next time you give the talk.
%
%\end{myitemize}
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%
%\begin{myitemize}
%\item  If you use typeset slides you may have the option of projecting directly from a computer through a projection device, as opposed to using transparencies on an overhead projector. 
%
%\item  One advantage of working from the computer is that you can change the slides right until the last minute. Another is that programs such as Microsoft PowerPoint permit sophisticated techniques to be used, such as animation and building up a slide layer by layer. There are several disadvantages:
%
%\item  The quality of computer projection systems is variable. Unless a sharp, bright image is produced, it will be better to use transparencies.
%
%\item  There is a tendency for authors to emphasize presentation at the expense of content when using the fancier presentation programs. One seminar organizer I know has banned the use of Power Point presentations.
%
%\item  More things can (and do) go wrong so, as a golden rule, have printed slides ready just in case!
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%\end{myitemize}
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%\begin{myitemize}
%\item In this section I give some examples of how to improve slides. 
%
%\item  In each case two slides are shown: an original draft and a revised version. 
%
%\item  Keep in mind that the slides would be printed in much larger type, and magnified further by the projector. 
%
%\item  Therefore, what appear here to be only minor flaws will be much more noticeable in practice.
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 \begin{figure}
 \centering
 \includegraphics[height=0.7\textheight, width=0.7\textwidth]{pic/figure-10-2-slides-1.png}
% \caption{ }
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 \begin{figure}
 \centering
 \includegraphics[height=0.7\textheight, width=0.7\textwidth]{pic/figure-10-2-slides-2.png}
% \caption{ }
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\begin{myitemize}
\item  The slide gives some numerical results. {\color{red}The second version improves on the first in several ways.}

    \begin{myitemize}
    \item  The definition of the matrix has been changed from {\color{red}a precise but cryptic form} to a descriptive one.
    \item The quantities $\rho^N$, $\rho^R$, and $\gamma_2$ would have been defined on an earlier slide, but the audience may need to be reminded of what they represent. Therefore, {\color{red}a description in words} has been added.
    \item Since only the order of magnitude of the error is relevant, the numbers are quoted to just one significant figure. This reduces the {\color{red}visual complexity} of the table. To reduce the complexity further, the column headed $\rho^C(\hat{y})$ has been removed, and the minimum number of rules has been used in the table.
    \end{myitemize}

\end{myitemize}

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%
%\begin{myitemize}
%\item In Figure 10.3, the original slide contains a theorem as it would be stated in a paper. The revised slide omits some of the conditions, which can be mentioned briefly as the slide is explained, and reminds the viewer of the definition of $\kappa_2(A)$. The simplified slide is much easier to assimilate.
%
%The revised slide in Figure 10.5 omits some unnecessary words and symbols from the slide in Figure 10.4. The slide in Figure 10.6 has far too many words. Ruthless pruning produces the much more acceptable version in Figure 10.7; as usual, the information removed from the slide would be conveyed when giving the talk. 
%
%Note that the parentheses in the expressions for $\beta_{n+1}$ and $\pi_{n+1}$ were unnecessary and so have been omitted, and the final error bound is written in a more readable way. 
%
%Another possible improvement is to remove the definitions of $\alpha_0$, $\beta_0$ and $\pi_0$, which may be irrelevant for this slide.
%
%\end{myitemize}
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%\begin{myitemize}
%\item See Section 11.3 in the next chapter.
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{\Large\color{blue}
%《数学写作》第七章: Revising a Draft
Mathematical Writing 

\vspace{0.2cm}

Chapter 11: Giving a Talk
}
\vspace{1cm}

NJH

\vspace{1cm}

November 26, 2020

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\begin{frame}{Contents }

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\begin{myenumerate}
\item 11.a. Quotations
\item 11.1. Preparation
\item 11.2. Delivery
%\item 11.3. Further Reading

%\item 
\end{myenumerate}

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\begin{frame}{11.a.1. Quotations - 1 }

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\begin{myitemize}
\item You are now ready to start your speech ... \\
Do not thank the Chairman or thank the audience.\\
Do not attempt to tell them that you are pleased to be there, \\
and above all do not elaborate on your lack of skill as a public speaker. \\
{\color{red}Audiences do not like speakers to apologise.} \\
They find it embarrassing.

\hfill {\footnotesize - Peter Kenny}\\
 %- PETER KENNY, 
\hfill {\footnotesize A Handbook of Public Speaking for Scientists and Engineers (1982)}


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\begin{frame}{11.a.2. Quotations - 2 }

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\item  {\color{red}The utterance should} not be rapid and hurried and consequently unintelligible but {\color{red}slow and deliberate conveying ideas with ease} from the Lecturer and infusing them with clearness and readiness into the minds of the audience. 

\hfill {\footnotesize - Michael Faraday, Letter to B. Abbott (11 June 1813) }
%- MICHAEL FARADAY, Letter to B. Abbott (11 June 1813)

%\item A Lecturer falls deeply beneath the dignity of his character when he descends so low as to angle for claps and asks for commendation yet have I seen a lecturer even at this point. 
%
% - MICHAEL FARADAY, Letter to B. Abbott (18 June 1813)

%\item  The Three Rules of Public Speaking: \\
%Be forthright. \\
%Be brief. \\
%Be seated.
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% - SUSAN DRESSEL AND JOE CHEW, Authenticity Beats Eloquence (1987)

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\begin{frame}{11.1.1. Preparation - 1 }

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\begin{myitemize}
\item  The previous chapter described how to design a talk and prepare the slides. Now I turn to the process of giving the talk.

\item  {\color{red}Having written the slides, you need to decide what you want to say while each slide is displayed.} You should explain and amplify what is on the slides rather than simply read them word for word, and add anecdotes, stories and humour as you feel appropriate. 

\item  {\color{red}If you are an inexperienced speaker you should find it helpful to begin by writing the talk out in full sentences, taking care to include everything that you want to say.} This text should not be used for the actual presentation, since reading from a prepared text always sounds unnatural and stilted, but should be the basis for preparing notes from which you work. 

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\begin{frame}{11.1.2. Preparation - 2  }

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\begin{myitemize}
\item {\color{red}I like to write on the backing paper for each slide the key words and phrases that I will use to introduce and explain the slide.} I use these notes when practising, but I find that I usually know them by heart by the time of the talk. If I give the talk again a few weeks or months later, the notes save me a lot of preparation time. 

%\item  The notes can alternatively be written onto prompt slides inserted between the real slides and numbered specially (la, 2a, etc.). A few of the prompt slides can even be displayed on the projector if the occasion demands it. Whatever approach you use, try to use the initial period of planning to make your language more vivid by finding synonyms for frequently occurring words and developing colourful imagery.

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%\begin{myitemize}
\item {\color{red}Most experienced speakers are happy if the audience go away having understood and remembered one point or idea from the talk.} You should therefore try to emphasize just one or two key points, repeating them in different words and at different points in the talk, to ensure that they are taken in.

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\begin{myitemize}
\item  {\color{red}Make sure you know how to pronounce all the technical terms and names that you will use in the talk}, since mispronunciation can be embarrassing for both you and the audience.

\item  {\color{red}Obviously you should practise the talk}, but how much practice is needed varies from person to person and also with experience. I need to practise my talk, speaking aloud, two or three times before I am happy with it, but I find that too much practice can make the talk too slick and lacking in spontaneity. 

\item  {\color{red}It is important to time each practice talk}, so that you can adjust the length to avoid overrunning or finishing very early. For an inexperienced speaker, it is very helpful to give a practice talk in front of friends and colleagues, especially if they can be persuaded to offer constructive criticism.

%\item  Needless to say, it is vital to ensure that your slides have been correctly prepared and that you do not lose them on the way to the venue. This way, you should avoid the embarrassment of the professor who began his talk at a major conference only to realise that his secretary had photocopied onto the paper backing sheets and not the slides themselves!


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%On the day of the talk or earlier, I strongly recommend going in advance to the room where the talk will be given and investigating the following questions.
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%\item  How big is the room? How loud will you have to speak to be heard? (Remember that a voice sounds louder in an empty room, because bodies absorb sound.) Will a microphone be used? If so, it will almost certainly be a clip-on type, possibly with a cigarette packet-sized unit that you must put in a pocket or clip onto a belt; make sure you are suitably attired. 
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%\item  A hand-held microphone, occasionally encountered, is difficult to hold at a constant distance from the mouth while you move around, and makes it difficult to talk while you change slides.
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%\item What pointing strategy will you use: to point at the screen or at the slide on the projector? The screen may be too big or far away to point at with a traditional metal or wooden pointer, though a laser pointer may still be usable. 
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%\item  Laser pointers are difficult to use effectively: they need to be held very steady to avoid shake and the unpredictable, short-lived appearances of the (usually red) dot at different points on the slide can irritate the audience. 
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%\item  If you do point at the screen, take care not to talk into the screen, or your voice will not project into the audience. If you point at the slide on the projector use a pen or pointer rather than a finger, and lay it on the projector if possible to avoid shake.
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%\item  Where will you put your pile of slides? Usually, there is a suitable table close to the projector. The location of the table and the projector will determine where you stand as you give the talk. Consider whether it is possible to move them to a better position. Try not to walk across the projector's image, though this is often unavoidable if you are using two projectors.
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%\item  Is the projector working and do you know how to use it? Find the on/off switch for the projector (sometimes a nontrivial task!). Put a slide on the projector and check that the projector is correctly aligned and sharply focused. Check if there is a spare bulb and how it is accessed. 
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%\item  Find a place to stand where you will not block the audience's view of the screen. If there are two projectors, compare the clarity and size of their images, and if one is clearly superior consider using it as the master projector rather than alternating between the two. Put a transparency on the projector and see if it slips; if so, it will need holding in place with a coin or eraser (or the projector can be propped up with a folded piece of paper under its legs).
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%\item  Where are the light switches? The chairperson should take care of extinguishing the lights near the projector screen, but you might have to step in if he or she is unprepared.
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%\item  Where is the chalk (if there is a blackboard), or where are the pens (if there is a whiteboard)? You may want to write on the board to explain something more fully, or in answer to a question. You may need to correct or add to a slide, so have your pens handy.
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%\begin{myitemize}
%\item  This reconnaissance not only prepares you for most eventualities, but makes you feel more confident in the run-up to the talk because you know that you are prepared.
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%\item  On the day of the talk, if you are not the first speaker you have an advantage in that you can observe the other speakers and note and avoid any mistakes they make in their use of the equipment and the stage.
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%\item  Just prior to the talk, it may help to take a few minutes of "quiet time" (if possible), in which you focus on the main points you wish to convey and on how you will deliver the talk.
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\begin{frame}{11.2.1. Delivery - 1 }

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\begin{myitemize}
\item {\color{red}Try to give a dynamic presentation that conveys enthusiasm for your subject.} The total amount of time that you are taking from the audience (size of audience x length of talk) may amount to many hours, so you should make every effort to give an inspiring and professional presentation.

\item {\color{red}Two of the most common and easily avoided mistakes are to speak too quickly and to speak too quietly.} Most people can be heard at the back of the room provided they face the audience and remember to speak more loudly than usual. 

%\item Asking "Can you hear me at the back?" should not be necessary; it should be possible to judge from the sound of your voice and the faces of the audience whether they can hear. 

\item {\color{red}You should speak more slowly than in normal conversation}: Kenny suggests not more than 100 words per minute, while Calnan and Barabas suggest an upper limit of 120 words per minute. 


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\begin{myitemize}
\item {\color{red}Following this advice is not easy because nerves tend to cause us to speak more quickly than usual.} Moreover, when in full flow of explanation we may feel we have so much to say that we must rush to fit it all in; practising beforehand should preclude this feeling by giving confidence in the timing of the talk. 

\item {\color{red}One of the reasons for speaking slowly, particularly in a large room, is that when we raise our voices the different types of sound scale at different rates.} For example, we cannot increase the volume of the consonants f, k, p and t - just that of the vowels around them. Therefore a slower rate of delivery is needed to keep our speech understandable. 

%\item Of course, it is also important to keep in mind that the audience may contain people whose first language is not English and for whom understanding spoken English presents difficulties.


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\begin{myitemize}
\item {\color{red}Avoid um's, ah's and other sounds designed to fill the gap between one sentence and another.} Pause instead. Pauses give the audience a moment to digest what you have just said and to anticipate what is coming next.
They give you time to sense the mood and reaction of the audience and to gather your thoughts. 

\item  {\color{red}Pausing while changing a slide gives the audience a chance to ask questions.}
% (if you do not want questions during the talk you can ask that they be held until the end). 
Note that a pause never seems as long to the audience as it does to the speaker.

\item  For variety, instead of stating a fact try posing it as a question and then answering it: {\color{blue}\it "Why doesn't it suffice to look at the eigenvalues? Because the matrix is highly nonnormal."} Anticipate the audience's line of thinking with phrases such as {\color{blue}\it"You may be wondering why ... ."} 
%The talk can be enlivened by the use of analogies, either from the technical field under discussion or from life in general.


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\begin{myitemize}
\item {\color{red}Look at the audience as much as you can, trying to cover the whole audience.} Eye contact is vital if you are to build up a rapport with the audience and maintain their interest. 

\item  If you stare into the screen or at the projector you will miss the feedback given by the faces in the audience. 

\item  {\color{red}Never turn your back on the audience}, even when pointing; a sideways stance allows you to look at the screen or the audience with small movements of the head.

\item  {\color{red}A tip I learned from a drama teacher is to exaggerate gestures made with the hands or arms}: pointing with a fully stretched arm is more effective than pointing with a hand.


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\begin{myitemize}
\item {\color{red}Nerves are perfectly natural and affect all speakers.} They can be employed to advantage because they generate the energy necessary for an effective talk. Taking a few deep breaths before beginning to speak helps to control nerves. Provided that you have prepared properly, nerves should disappear once the talk is underway. Remember that the audience are on your side and want you to succeed, so do not be daunted by them.

\item  Vary the pitch of your voice, mainly downwards, to maintain the listeners' interest. {\color{red}Speaking in a monotone is a sure way to send an audience to sleep.} Watch out for the tendency for nervousness to cause the voice to rise in pitch, because of tightening of the muscles around the throat and voice box. Variations in the speed and rhythm of your speech are also worth aiming for.


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\begin{myitemize}
\item {\color{red}Finishing on time is important.} Overrunning indicates a lack of professionalism and is apt to lose you the audience's attention, as they ponder the coffee break, the next talk or lunch. 

%\item  At a conference, exceeding your time is particularly discourteous as it either takes away time from the next speaker or causes the carefully worked out schedule to be upset (a particular nuisance if there are parallel sessions). 

%\item  I recommend writing the finishing time in your notes before you begin. It is easy to forget, and the simple mental calculation of when you should stop can be difficult when you are speaking.

\item  {\color{red}Time seems to pass more quickly for the speaker than for the audience, which is one of the reasons why some speakers overrun.} To keep yourself aware of the time it is a good idea to remove your watch and place it on the table next to the slides. You can then regularly glance at the watch without the audience noticing.

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\begin{myitemize}
\item  {\color{red}Signal that the end is coming to awaken the interest of the audience and to increase the chance of your final message be remembered.} If the end of the talk is well constructed it should be obvious when you have spoken your last sentence, but you may want to add "thank you" or "thank you for your attention" , particularly if the chairperson seems to be dozing! %Let the chairperson ask if there are any questions and select the questioners. 

\item  Immediately asking for questions yourself precludes applause and takes listeners' thoughts away from your concluding sentences. {\color{red}In a large room, repeat a question on receiving it. This ensures that all the audience hear the question and gives you time to think of an answer.} 

%\item  If you are not sure how to answer the question, it is perfectly acceptable to say something like "That's a very interesting question to which I don't know the answer" , perhaps adding "Can we talk about it afterwards?"
%One word answers "Yes" or "No" are acceptable, and allow time for further questions. Do not give answers so long that they sound like a second lecture.

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\item It is tempting to relax at the end of your talk, which can cause you to lose concentration during the question and answer session. {\color{red}However, you need to be fully alert to answer difficult or unexpected questions, and if you handle questions poorly you will leave the audience with a negative impression of your work.}

\item  You may like to prime one or two friends with questions, as one question is usually all it takes to break the ice and start a stream of questions. 

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%\item  Some more mistakes to avoid. 
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%%\item  Don't put a sweaty palm on the slide. It will produce a blotch on the screen that gradually fades before the audience's eyes.
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%%\item Avoid blocking the path of light from projector to the screen as well as the audience's view of the screen. I remember one speaker being asked by a member of the audience "Can you pace nervously back and forth so that I can see the screen?"
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%%\item  Avoid noisy slide management. If the slides have backing sheets, tear them off before the talk (if you don't, the act of tearing off a backing sheet does at least create a necessary pause). If you keep the slides in a ring binder, don't open and close it between each pair of slides.
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%%\item  Don't leave the screen blank for more than a few seconds, as this may distract and dazzle the audience. Turn the projector off or put a piece of paper on the projector.
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\begin{frame}{11.2.9. Delivery - 9 }

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\begin{myitemize}
\item  %Figure 11.1 summarizes advice on preparing and delivering a talk. Finally, 
{\color{red}It is important to appreciate the role that talks play in a research community.} 
    \begin{myitemize}
    \item  They are the focus of the community's interactions and a celebration of its achievements and they bring the subject to life. 
    \item  Talks provide an opportunity to gain insight into a researcher's work and personality that cannot be obtained from the printed page. 
    \item  Every subject has its star speakers who are renowned for the quality of their talks. 
    \item  You can learn a lot about your subject and about giving talks by listening to these people.
    \end{myitemize}
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\begin{frame}{11.2.10. The Ten Commandments of Giving a Talk }

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\begin{myitemize}
\item 1. Design the talk for the audience.
\item 2. Prepare thoroughly and rehearse the talk.
\item 3 . Produce clear, legible slides.
\item 4. Arrive early and check the lecture room.
\item 5. Speak slowly and loudly.
\item 6. Be enthusiastic about what you say.
\item 7. Look at the audience as you speak.
\item 8. Don't fidget with the slides or the pointer.
\item 9. Finish on time, or early.
\item 10. Answer questions courteously, concisely, and honestly. %and admit it if you don't know the answer.
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%\item  Excellent advice on both writing and giving a talk can be found in various places. 
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%\item  In {\color{blue}\it How to Talk Mathematics}, Halmos addresses advice to the young mathematician, but what he says about such topics as simplicity, detail, organization and preparation should be read by all mathematicians. 
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%%\item  Forsythe gives suggestions to students on how to present a talk about a mathematical paper. 
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%\item  In {\color{blue}\it A Handbook of Public Speaking for Scientists and Engineers}, Kenny discusses all aspects of public speaking for the scientist, including preparation of material, choice of visual aids, and presentation. 
%He recommends tongue twisters for the busy scientist to improve articulation!
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%\item There are many general books on public speaking and much of what they say is relevant to scientific talks. 
%One of my favourites is Aslett's {\color{blue}\it Is There a Speech Inside You?}. 
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%\item  The chapter "Speaking at Scientific Meetings" in Booth's Communicating in Science [36] is full of good advice. 
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%\item  Turk's Effective Speaking [279] is a general guide to speaking; it reports research findings on topics such as audience psychology, memory and non-verbal communication. 
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%\item  Calnan and Barabas [50] have written an excellent guide to Speaking at Medical Meetings, most of which applies to any form of technical speaking.
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%\item  Byrne [48] and Garver [102] both offer practical advice on preparing and giving technical presentations. 
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%\item A transcript of a (presumably imaginary) talk that breaks so many rules that it is funny is given in "Next Slide Please" [10]. 
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%\item The American Statistical Association gave workshops on improving technical presentations at its 1980-1982 annual meetings and the material that emerged from the workshops is summarized in "Presenting Statistical Papers" [86]. 
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%\item  Common mistakes in using an overhead projector are discussed by Gould in "The Overhead Projector" [113], and in "Visual Aids-How to Make Them Positively Legible" [114] he discusses the choice of letter size for slides, pointing out the effects on legibility of room layout and screen size and positioning. 
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%\item There are many general- books on public speaking and much of what they say is relevant to scientific talks. One of my favourites is Aslett's Is There a Speech Inside You? [11].
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%\item  An aspect of public speaking that is often overlooked is proper use of the voice. Two useful books in this respect are Berry's Your Voice and How to Use It Successfully [29] and Rodenburg's The Right to Speak [237], both of which contain advice on pronunciation, breathing and relaxation.
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%\item  Memory systems can help you to remember your notes, or facts and figures not shown on the slides. See any of the books by Buzan or Lorayne, including [46], [183], [184].
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\begin{myitemize}
\item This is \S 2 from {\color{blue}\it Mathematical Writing} by Donald E. Knuth. 

\item In the following excerpt from a term paper, 
    \begin{myitemize}
    \item $N$ denotes the nonnegative integers, $N^n$ denotes the set of $n$-tuples of nonnegative integers, and $A_n = \{(a_1,\cdots,a_n) \in N^n \mid a_1 \ge \cdots \ge a_n \}$.
    \item If $C,P \subset N^n$, then $L(C,P)$ is defined to be $\{c+p_1+\cdots+p_m \mid c\in C, m \ge 0,\textrm{ and }p_j\in P
    \textrm{ for } 1 \le j \le m\}$. 
    \item We want to prove that $L(C,P) \subseteq A_n$ implies $C,P \subseteq A_n$.
    \end{myitemize}

\item The following proof, directly quoted from a sophomore term paper, is {\color{red}\bf mathematically correct} (except for a minor slip) but {\color{red}\bf stylistically atrocious}:

\end{myitemize}

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\begin{frame}{K.2.2. First Solution }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

{\small

$L(C,P) \subset A_n$

$C\subset L \Longrightarrow C \subset A_n$

Spse $p \in P, p \notin An \Longrightarrow p_i < p_j$ for $i < j$

$c + p \in L \subset A_n$

$\therefore c_i + p_i \ge c_j + p_j$ but $c_i \ge c_j \ge 0, p_j \ge p_i$ $\therefore (c_i - c_j) \ge (p_j - p_i)$

but $\exists$ a constant $k \ni c + kp \notin A_n$

let $k = (c_i - c_j)+1$ \hspace{0.5cm} $c + kp \in L \subset A_n$

$\therefore c_i + kp_i \ge c_j + kp_j \Longrightarrow (c_i - c_j) \ge k(p_j - p_i)$

$\Longrightarrow k-1 \ge k\cdot m$ \hspace{0.5cm} $k,m \ge 1$ \hspace{0.5cm} Contradiction

$\therefore p \in A_n$

$\therefore L(C,P) \subset A_n \Longrightarrow C,P \subset A_n$ and the

lemma is true.

}

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\begin{frame}{K.2.3. Second Solution - Part 1}

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%每页详细内容

\begin{myitemize}
\item A possible way to improve the quality of the writing:

\vspace{-0.1cm}

\begin{center}
\fbox{\begin{minipage}{13.5cm}
Let $N$ denote the set of nonnegative integers, and let
\begin{eqnarray*}
N^n = \{(b_1,\cdots,b_n) \mid b_i\in N \textrm{ for } 1 \le i \le n\}
\end{eqnarray*}
be the set of $n$-dimensional vectors with nonnegative integer components.

We shall be especially interested in the subset of ``nonincreasing'' vectors,
\begin{eqnarray}
A_n = \{(a_1,\cdots,a_n) \in N^n \mid a_1 \ge \cdots \ge a_n \}
\end{eqnarray}

\end{minipage}}
\end{center}


\end{myitemize}

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\begin{frame}{K.2.4. Second Solution - Part 2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{center}
\fbox{\begin{minipage}{13.5cm}

If $C$ and $P$ are subsets of $N^n$, let
\begin{eqnarray}
L(C,P) = \{c + p_1 + \cdots + p_m \mid c\in C,m\ge 0, \textrm{ and } p_j \in P \textrm{ for } 1 \le j \le m\}
\label{LCP}
\end{eqnarray}
be the smallest subset of $N^n$ that contains $C$ and is closed under the addition of elements of $P$.

\end{minipage}}
\end{center}


\begin{center}
\fbox{\begin{minipage}{13.5cm}

Since $A_n$ is closed under addition, $L(C,P)$ will be a subset of $A_n$ whenever $C$ and $P$ are both contained in $A_n$. We can also prove the converse of this statement.

\end{minipage}}
\end{center}


\begin{center}
\fbox{\begin{minipage}{13.5cm}

Lemma 1. If $L(C,P) \subseteq A_n$ and $C\neq\emptyset$, then $C\subseteq A_n$ and $P \subseteq A_n$.

Proof. (Now it's your turn to write it up beautifully.)

\end{minipage}}
\end{center}

%\end{myitemize}

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\begin{frame}{K.2.5. Third Solution - Part 1  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}
\item Here is one way to complete the exercise in the previous section.

\item (But please try to work it yourself before reading this.)

\item Note that a few clauses have been inserted to help keep the reader synchronized with the current goals and subgoals and strategies of the proof.

\item Furthermore the notation $(b_1,\cdots,b_n)$ is used instead of $(p_1,\cdots,p_n)$, in the second paragraph below, to avoid confusion with formula (\ref{LCP}).

\end{myitemize}

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\begin{frame}{K.2.6. Third Solution - Part 2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{center}
\fbox{\begin{minipage}{13.5cm}

{\footnotesize
{\color{blue}Proof.} Assume that $L(C,P) \subseteq A_n$. Since $C$ is always contained in $L(C,P)$, we must have $C \subseteq A_n$; therefore only the condition $P\subseteq A_n$ needs to be verified.
If $P$ is not contained in $A_n$, there must be a vector $(b_1,\cdots,b_n) \in P$ such that $b_i < b_j$ for some $i<j$. We want to show that this leads to a contradiction.
Since the set $C$ is nonempty, it contains some element $(c_1,\cdots,c_n)$.
We know that the components of this vector satisfy $c_1 \ge \cdots \ge c_n$, because $C \subseteq A_n$.

Now $(c_1,\cdots,c_n) + k(b_1,\cdots,b_n)$ is an element of $L(C,P)$ for all $k\ge 0$, and by hypothesis it must therefore be an element of $A_n$. But if we take $k = c_i - c_j + 1$, we have $k \ge 1$ and
\begin{eqnarray*}
c_i+kb_i \ge c_j+kb_j,
\end{eqnarray*}

hence
\begin{eqnarray}
c_i - c_j \ge k(b_j-b_i).
\end{eqnarray}

This is impossible, since $c_i-c_j=k-1$ is less than $k$, yet $b_j-b_i\ge 1$.
It follows that $(b_1,\cdots,b_n)$ must be an element of $A_n$. \hfill $\blacksquare$
}

\end{minipage}}
\end{center}

%Note that the hypothesis $C \neq \emptyset$ is necessary in Lemma 1, for if $C$ is empty the set $L(C,P)$ is also empty regardless of $P$. [This was the ``minor slip.'']

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\begin{frame}{K.2.7. Fourth Solution - Part 1 }

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%每页详细内容

\begin{myitemize}
\item BUT ... don't always use the first idea you think of. 

\item The proof above actually commits another sin against mathematical exposition, namely the unnecessary use of proof by contradiction. 

\item It would have been better to use a direct proof.

\item This form of the proof has other virtues too: It doesn't assume that the $b_i$'s are integer-valued, and it doesn't require stating that $c_1\ge \cdots \ge c_n$.
\end{myitemize}

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\begin{frame}{K.2.8. Fourth Solution - Part 2  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{center}
\fbox{\begin{minipage}{13.5cm}

{\color{blue}Proof.}
Let $(b_1,\cdots,b_n)$ be an arbitrary element of $P$, and let $i$ and $j$ be fixed subscripts with $i<j$;
we wish to prove that $b_i\ge b_j$. Since $C$ is nonempty, it contains some element $(c_1,\cdots,c_n)$.
Now the vector $(c_1,\cdots,c_n) + k(b_1,\cdots,b_n)$ is an element of $L(C,P)$ for all $k \ge 0$, and by hypothesis it must therefore be an element of $A_n$. But this means that $c_i +kb_i \ge c_j +kb_j$, i.e.,
\begin{equation}%\tag{3}
c_i-c_j \ge k(b_j-b_i),
\end{equation}
for arbitrarily large $k$. Consequently $b_j-b_i$ must be zero or negative. We have proved that $b_j-b_i\le 0$ for all $i<j$, so the vector $(b_1,\cdots,b_n)$ must be an element of $An$. \hfill $\blacksquare$

\end{minipage}}
\end{center}


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